That’s a Moiré

I was sitting in the back seat of my parents’ car, stopped at a junction. I could see two cars in front of ours and both had their indicators flashing. The amber lights were out of phase: one turned on just as the other went off, but then, over the course of a few minutes, they teamed up and were flashing in unison. Then, a few minutes later, they returned to being out of phase.

That memory has always stayed with me. I was a young child, and it would be years before I studied physics and learned about wave oscillations and beat frequency—the technical term for the phenomenon I had witnessed. The flashing frequencies of the cars’ indicator lights were slightly different, causing them to slowly drift in and out of phase with one another. The beat frequency is the frequency with which this phase oscillates, and as we’ll see below, it is equal to the difference between the two flashing frequencies. Continue reading


On the cover: Vhat? Vhere? Venn

More often than I care to admit, I find myself sitting in the audience of a maths lecture or seminar completely and utterly lost as to what the speaker is going on about. What are they talking about? How does this relate to stuff I know about? Where does this fit within the sphere of mathematics as a whole? In fact, most of the time I am lost beyond the first slide of a presentation. In an endeavour to minimise the possibility that audience members would experience this feeling that I know all too well, I recently introduced myself at the start of a talk with this slide:

You could be forgiven for remarking “My, what a beautiful Venn diagram you have there!” Indeed, I too was under the impression that what I had created was in fact a Venn diagram. Continue reading


In conversation with Dominique Sleet

There are many qualities attributed to mathematicians that we can be proud of: we’re logical, meticulous, intelligent, even creative. Despite maths being revered for needing all of these excellent attributes, one thing we are perhaps not so renowned for is our communication skills: most of the world still finds maths intimidating and opaque. That’s why Chalkdust sat down with science communication expert Dominique Sleet to learn the secrets that will help us to share the beauty of maths as far and wide as possible.

Science explained

Dominique began her science communication life as an explainer at the Science Museum in London. Many London-based Chalkdust readers will be keenly familiar with the Science Museum, but for those who are not, the Science Museum is pretty much exactly what it says on the tin. As well as many more traditional static galleries, it has several interactive galleries aimed at young people, which are like science-themed playgrounds.

The Science Museum (Wikimedia Commons user Shadowssettle, CC BY-SA 4.0)

Explainers are tasked with (you guessed it) explaining the science behind what they’re doing. “You get to play with some fun things, like putting flowers into liquid nitrogen and smashing them or blowing up a hydrogen balloon. The whole museum ethos is to build an association between fun and science. If there’s some learning in there then that’s great, but there doesn’t have to be; it’s just about nurturing that relationship.” But there’s plenty to learn if you’re looking. “Some of the science behind the exhibits is beautiful. We had this exhibit where you would look through polarising lenses at a thin layer of ice and you could see all these feathery beautiful patterns with amazing colours. I’m a nerd, I like science.”

Those of us who grew up to be Chalkdust editors could spend a cheerful afternoon churning out algebra, but for most children the liquid nitrogen has the more evident appeal. “Maths has its challenges. It’s a lot more abstract. With science, as long as you use the right language, you can make almost anything accessible, whereas with maths, you often need to have prior knowledge.” And don’t forget that intimidation. “People have this barrier, it’s almost like a badge, ‘I don’t do maths.’ And it’s really sad. So before you’ve even begun, you have to overcome this preconception of maths being like an alien language, and only for clever people.” People are often more receptive to the content if they don’t know that it’s maths—”Pattern Pod was my favourite gallery, which was for under-eights and all about maths, but we were looking at patterns and didn’t label it as maths.” Unfortunately, the plausible deniability can’t last forever, and inevitably your audience will notice that they are being subjected to maths: what then? “You need to show why something’s important and make it relevant to everyday life. You can’t get into the depth that you’d need to understand all the maths behind the exhibits, but you can go into some detail about how the maths was used, how it changed the world, and what impact it had on people.” The work does not end there however. How do we get people to turn up for maths in the first place?

A royal invitation

Dominique talking in the Royal Institution lecture theatre

Receiving millions of visitors per year, the Science Museum is well-placed to reach out to people who wouldn’t usually be interested. “But even then there are barriers. I remember doing an outreach programme in south-east London and people hadn’t even heard of the Science Museum. That’s why outreach is really important. Going out into local communities and finding the people where they are in their everyday lives.”

Dominique’s next job was at the Royal Institution (also in London), where she worked on everything from their famous annual Christmas lectures to their extensive year-round masterclass programme. So what’s the trick to running a maths masterclass? “Pick a topic that interests you because if you’re passionate about the topic then you’re halfway there. The kids aren’t going to be excited about something if you’re not excited yourself. In the same breath, you need to realise that, while you might find something amazing, other people really don’t. You’ve got to show them why it’s interesting.”

A braid made during one of Matthew Scroggs’s RI masterclasses

It was nice to hear a shoutout for one of Chalkdust‘s own, who apparently is quite the master of masterclasses himself. “This will sound like I’m sucking up, but I remember a session with Matthew Scroggs getting the kids to explore different braiding patterns. There’s actually some really interesting maths going on, because some combinations of braid would work and some of them wouldn’t. But at the end he was saying he doesn’t really understand it, he doesn’t know what makes a good braid and what doesn’t.” This open-ended aspect of maths often isn’t apparent until university, and school often leaves people with the idea that all the maths has been done. “Kids have this idea that maths can only be right or wrong, but in fact there can be lots of exploration. And maths can be really quite creative.”

When Dominique learned that the 2019 Christmas lectures would be focused on maths, and feature veritable maths celebs Hannah Fry and Matt Parker, she jumped at the chance to be involved. “My role was Christmas lectures assistant, a kind of catch-all. On the night itself, I’m the one in the front row, on the laptop with little prompts for Hannah—and at the same time, trying to keep an eye on messages from livestreaming venues, to make sure they’re all happy.” And it was a lot of hard work. “Very high pressure and some of the team would work until two o’clock in morning. It was crazy.” One of her contributions turned out to be very prescient, in a segment designed to show how mass vaccination succeeds. “I suggested that we use surgical masks to indicate that the kids were vaccinated and that they can’t catch the virus. Now looking back on it—oh my God! Told the future!”

Dominique with Christmas lecturer Hannah Fry

Of course, communicating maths for TV brings with it some new challenges. “Sometimes there will be conflicting priorities between the production team and the Royal Institution. The production team want everything to look flashy. Whereas obviously the RI still want it to be interesting, but we also want to make sure the integrity of the maths is still there.” Those who watched the Christmas lectures (if you didn’t, you should hang up your maths fanatic hat right now) will recall a specific sequence which involved schoolchildren lining up and then taking a step to either the left or the right based on the result of a coin toss. “What we were trying to show was that probabilities can help you predict outcomes. So we wanted to get this lovely bell-shaped curve from all the students moving about, but we didn’t get what we were hoping for. Partly because it’s hard to instruct a large group of people to do exactly what you’re asking, but also simply because probability doesn’t offer any guarantees.” So where did that leave the narrative of the lecture? “The fact that the kids didn’t do what we thought is actually a really interesting point in itself. But from a TV perspective, that’s the opening demonstration and we can’t go off on a tangent. So we ended up having a montage of two different schools in the lecture.”

Widening participation

As Dominique moves on to her next job working on the outreach programme at Imperial College London, she finds that the university setting has an increased focus on widening participation—so how do we convince young people from underrepresented groups to consider maths? She says an obvious start is making your event free if possible (since financial barriers often have a big impact), and ensuring diverse role models are present. “Representation does matter—look for different people from different backgrounds, from different areas, as well as different topics.” Marketing is also crucial. “You can have the best outreach in the world, but if nobody knows about it and it’s not reaching the right people, then it’s not doing anything.” But don’t think your job is over once you have them in the room. “The audience should be kept at the forefront of your mind. You need to be thinking about who your activity is for, what you want them to learn and how are you going to make sure they actually understand?”

Finally, she encourages everyone to take the necessary time and effort to accommodate accessibility needs. “Putting a bit of effort in to make it as accessible as possible, whether that’s looking at the colour scheme you’re using for colour blindness, not putting too many words on a screen, or having materials available in advance or in large print. All of those accessibility things can feel like extra work but they’re only ever going to improve it. Good for everyone—not just the person you are trying to accommodate.”

Having only been at Imperial for three months, she is is still reasonably new to the job, but has a lot of positive things to say about what she has seen. Her current project focuses on sixth formers. “I think the programme itself is really worthwhile, it’s very intense. There are online courses, with mentoring sessions in small groups throughout the year as well as large welcome and closing events on campus, although as you can imagine these have had to convert to online events in recent times.” Looking to the future, she says: “There is a move to intervene earlier in children’s maths education so Imperial, like many other organisations, have a growing number of outreach programmes aimed at younger students.” We wish her all the best in her latest endeavour.


An odd card trick

Martin Gardner—one of history’s most prolific maths popularisers—frequently examined the connection between mathematics and magic, commonly looking at tricks using standard playing cards. He often discussed ‘self-working’ illusions that function in a strictly mechanical way, without any reliance on sleight of hand, card counting, pre-arrangement, marking, or key-carding of the deck. One of the more interesting specimens in this genre is a matching trick called the magic separation.

This trick can be performed with 20 cards. Ten of the cards are turned face-up, with the deck then shuffled thoroughly by both the performer and, importantly, the spectator. The performer then deals 10 cards to the spectator and keeps the remainder for herself. This can be done blindfolded to preclude tracking or counting. Not knowing the distribution of cards, our performer announces she will rearrange her own cards ‘magically’ so that the number of face-ups she holds matches the number of face-ups the spectator has. When cards are displayed, the counts do indeed match. She easily repeats the feat for hecklers who claim luck.

How the illusion works

The magic separation was first devised by Bob Hummer in his pamphlet Half-a-Dozen Hummers back in 1940, and since has been the subject of discussion in books by senior conjurors such as Bill Simon in Mathematical Magic from 1964, Karl Fulves in Bob Hummer’s Collected Secrets in 1980, and by the master of mathematical puzzles Martin Gardner in his aptly named book The Scientific American Book of Mathematical Puzzles and Diversions from 1959. The unusual step of allowing the deck to be out of the performer’s control makes the trick particularly baffling, but this is simply a diversion to further obscure its deterministic workings. What might these workings be?

The water and wine riddle: a litre of wine is transferred to the water container, then a litre of mixture is transferred back. Is there more wine in the water than water in the wine?

Gardner explained the trick in terms of a conservation of mass property, borrowing from the classic riddle of equal containers of water and wine. Namely, if a litre of wine is transferred to the water container, which is then stirred, and a litre of the mixture is subsequently transferred back to the wine container, is there more wine in the water than water in the wine?

Intuition often suggests there is, since the first transfer was pure wine, while the second, reverse transfer was a water and wine mixture. This intuitive, but incorrect supposition is what renders the magic separation trick so likewise remarkable. In fact, there is as much water in wine as wine in water: since the starting and ending volumes in the containers are equal, whatever wine is in the water container must be matched by whatever water is in the wine container.

This conservation principle can be concretely demonstrated for the magic separation by starting with two separate piles, one of 10 face-up cards and the other of 10 face-down cards. Now, make a one-for-one exchange of cards between the piles any number of times you wish. You can make each selection randomly and you can even shuffle each pile after each transfer. The only condition is that transferred cards maintain their original orientations. At the conclusion, each pile still has 10 cards that will now be some mixture of face-ups and face-downs. What one necessarily finds is that the number of face-up cards in one pile (analogous to the wine) equals the number of face-down cards in the other pile (analogous to the water). The ‘magic’ part of the magic separation is that our performer simply—though with some theatrical flourish—turns her pile over to force matching numbers of cards.

In Hummer’s original 1940 pamphlet, he referred to this trick as “sure fire” and it is now easy to see how this is the case. Indeed, the deterministic outcome enabled by combining conservation with the ‘turn over’ manoeuvre and the elegant misdirection of spectator shuffling has made the magic separation a standard part of the close-up magic toolkit. The original trick has since expanded into a large family of variations, both stylistically and in terms of the numbers of cards used.

The version above—10 cards up and 10 cards down—is popular and we shall call this a 10/10 configuration. Some other versions split a full deck evenly, while others use an uneven division of a deck, for example a 20/32 configuration. These variations go by many names within professional conjuring circles, including the match-upthe topsy turvy deck, and the gremlins.

Here come the mathematicians

The fact that there are so many versions of the magic separation and that they all seem to work on the same combination of conservation and turn over manoeuvre beckons a deeper mathematical look: is there some sort of general law that governs the magic separation family?

To start investigating this question, let $T$ be the total number of cards in the deck and $F$ be the number of face-up cards therein. Also, take $S$ as the total number of cards, face-up plus face-down, that the spectator is dealt. These are parameters that characterise a specific variation of the magic separation. However, separate from these are the variables, which, unlike parameters, are out of the performer’s control and which generally change each time the trick is performed because of shuffling. Following the Diaconis and Graham convention in their book Magical Mathematics, where an overbar indicates ‘face-up-ness’, let $\overline{\sigma}$ and $\sigma$ represent the respective numbers of face-up and face-down cards the spectator is holding and $\overline{\mu}$ and $\mu$ be the corresponding counts of the magician’s cards.

Four classes of cards, tallied respectively by $\overline{\sigma}$, $\sigma$, $\overline{\mu}$, and $\mu$, result from two types of division: spectator versus magician and face-up versus face-down.

The following tallies for cards held by each party are immediately implied: for the spectator $\overline{\sigma} + \sigma = S$ and for the magician $\overline{\mu} + \mu = T – S$. A third equation comes from noting that, although the cards are shuffled, the number of face-ups is fixed, no matter how the cards are distributed in any instance of the trick, meaning $\overline{\sigma} + \overline{\mu} = F$. The last, and most key ingredient is the observation that the number of face-up cards held by the spectator has to equal the number of initially face-down cards possessed by the magician because it is the conjuring ‘turn over’ manoeuvre that forces the matching effect into existence. In other words, the illusion only works if $\mu = \overline{\sigma}$.

These ingredients are now baked into the following mathematical pie. Rearrange the face-up equation as $\overline{\mu} = F – \overline{\sigma}$, substitute this into the equation tally for the magician’s cards, and then rearrange terms to find $\mu = \overline{\sigma} + T – S – F$. The only way $\mu = \overline{\sigma}$ can be satisfied is if the last three terms are self-cancelling, meaning that the mathematical condition enabling the magic separation is \[ F \>+\> S \>=\> T\>. \] For convenience, we will call this little result Hummer’s theorem.

Visualising the workings

Hummer’s theorem is actually quite profound in the sense that it explains how all extant variations of the magic separation work and also shows how to immediately invoke numerous newer versions, as limited, it seems, only by the number of cards the magician can physically handle. For instance, following some quick calculations using the theorem, one could readily perform much grander versions of the trick that combine, say, several full 52-card decks.

A Fisher table showing that a new drug is more effective than an old drug.

Another interesting aspect of Hummer’s theorem is that it immediately suggests how to visualise the workings of the magic separation at their most basic, mechanistic level by borrowing on $2\times 2$ contingency tables. Such tables are ordinarily used in the context of assessing whether two categorical variables, for instance medical treatment (old drug versus new drug) and clinical result (patient improves versus does not improve), are statistically related. One of the most common statistical calculations executed on such tables is something called Fisher’s exact test, so it will be convenient to call these $2\times 2$ structures Fisher tables.

Visualising the magic separation.

Now, in order to visualise the architecture of the magic separation, we use the columns for the people (spectator versus magician) and rows for the cards (face-up versus face-down). The diagram to the right shows the variables $\sigma$, $\overline{\sigma}$, $\mu$, and $\overline{\mu}$, and the parameters, $F$, $S$, and $T$, in such a table. A moment’s consideration should indicate that the only $2\times 2$ tables consistent with Hummer’s magic separation are those in which the leading diagonal entries are equal.

A 10/10 Hummer configuration (left), a 20/32 Hummer configuration (right).

A bit more consideration suggests that all Hummer-compliant tables are Fisher tables, but not all Fisher tables are Hummer tables. This is readily confirmed by example. The top two tables to the right satisfy diagonal equality, while the bottom table does not. Here, Hummer’s theorem is violated ($8+12\neq24$), so this instance lacks the required conservation property for the magic separation. These tables can be used to visualise the actual dynamics of the magic separation.

A non-Hummer table that violates the separation condition.

Tables representing a trick possess a conservation property: the margin totals are rigidly fixed. Because parameters $F$, $S$, and $T$ are likewise fixed for a particular variation of the magic separation trick, we can exploit the conservation property here to visualise the 10/10 pile experiment introduced above as a series of $2\times 2$ tables, one for each round of trades. The experiment starts with 10 face-ups for the spectator and 10 face-downs for the magician, resulting in the leftmost $2\times 2$ below.

In the first exchange, the spectator cedes a face-up card to the magician, who in turn passes back a face-down card, resulting in the round 1 table. Importantly, while the boxed variable tallies show each person’s new holdings, the margin tallies have not changed.

After shuffling both piles, the second iteration might see the same type of exchange, resulting in the table marked round 2. One could now step through many additional rounds of trades, sometimes exchanging cards of the same orientation, for which the table remains exactly the same, and sometimes exchanging cards of opposite orientation, whereby the boxed tallies are updated appropriately.

A performance of magic separation, represented as a pile experiment with no card exchanges made.

Every table would display constancy of margin totals and strict satisfaction of Hummer’s theorem, no matter how the cards are distributed. One possible endpoint to this 10/10 experiment is familiar from above.

In a sense, the magic separation trick itself could actually be thought of as a special case of this process that consists of the magician’s deal, with zero rounds of card exchange.

Heightened mystery of odd-numbered decks

In the original magic separation, Hummer emphasised that the deck size, $T$, must be an even number. Fulves’ instruction manual repeats this condition, as do Simon’s and another of Gardner’s books on mathematical magic. But here is a question: can the magic separation work if $T$ is odd?

The answer is not intuitively obvious. First, face-up cards must be matched one-for-one between spectator and performer, ie in pairs. Second, there are many tricks that do indeed require even-numbered decks (Diaconis and Graham discuss examples). However, since there were no parity conditions placed on $T$ in our derivation, we can immediately conclude that odd decks are indeed acceptable. A quick corollary to Hummer’s theorem evidently leads to a new and non-obvious extension of this 80-year-old trick!

Because this aspect is somewhat more difficult to perceive, a truth table showing all possible parity combinations can be helpful:

spectator magician $\overline{\sigma}+\overline{\mu}=F$ is satisfied?
case $F$ $S$ $\overline{\sigma}$ $\sigma$ $T-S$ $\overline{\mu}$ $\mu$
A even odd odd even even even even no
B even odd odd even even odd odd yes
C even odd even odd even even even yes
D even odd even odd even odd odd no
E odd even odd odd odd even odd yes
F odd even odd odd odd odd even no
G odd even even even odd even odd no
H odd even even even odd odd even yes

Half the cases—namely A, D, F, and G—are actually inaccessible because they violate $\overline{\sigma} + \overline{\mu} = F$. For instance, in case G, both people cannot hold an even number of face-ups if the total number of face-up cards in the deck, $F$, is odd.

For the admissible cases, B and C are the more pedestrian because the required parities are already established. Conversely, cases E and H are a little more interesting. Here, the magician’s own two subsets have different parities from one another and it is the conjuring ‘turn over’ that forces the parities of $\overline{\sigma}$ and $\mu$ to match because the latter now has become flipped. Hummer’s magic separation works even when there are an odd number of cards in a deck that itself starts with an odd number of face-up cards!

This ‘double-odd’ configuration is surprising and does not seem to have been empirically discovered over the eight decades that the magic separation has been performed.

KC Cole remarked in her book The Universe and the Teacup that mathematics can produce a “literal expansion of consciousness” and it seems that Hummer’s theorem, namely that the spectator’s card count plus the face-up card count must equal the size of the deck, marks yet another example of this amazing and long-established phenomenon. Keep that in mind the next time you dazzle your friends and neighbours with the magic separation.


Correction: Who is the best England manager?

Image: Кирилл Венедиктов, CC BY-SA 3.0.

Following the Uefa European football championship this summer, I have an update to my issue 13 article including Gareth Southgate’s performance as England manager.

In the England team’s entire history, it has only ever won 15 knockout games at major tournaments and Gareth Southgate was manager for 33.3% of those games.

England have made it to the semi-final stage of a major tournament five times (excluding the 1968 Euros, which consisted of four teams, so getting to the semi-final wasn’t much of an achievement). Southgate has been in charge for 40% of those.

England have reached the final of two major tournaments and Southgate is responsible for half of those. He is the best England manager ever (or at least in the top one, to paraphrase Brian Clough). In conclusion, I’d like to say…

Looking back on when we first met;
I cannot escape and I cannot forget,
Southgate, you’re the one; you still turn me on,
You can bring it home again.

(apologies to Atomic Kitten)


Pawns, puzzles, and proofs

I am sure you will agree that problem solving is an integral part of learning and creating new mathematics, but for many mathematicians recreational problem solving is also a favourite pastime. I am no exception to this; I have a fondness for riddles and brainteasers, an interest which later expanded to include chess problem solving. The amount one could write about chess problems could fill a book (in fact, many books have been written on the topic!). For today though, I want to introduce you to the world of one particular type of chess problem, namely retrograde problems. I am going to start by letting you in on a secret: retrograde chess problems are essentially logic puzzles set on a chessboard. And what’s more, they can be approached in the same way as typical maths problems and puzzles. Now I’ve really got your attention, haven’t I?

Back to square one

In retrograde problems one is typically given a chess position, possibly with some additional rules imposed by the composer. What do I mean by rules? They could tell you, for example, that the white king has not moved more than twice, or that pieces only moved to squares of the same colour they started. The composer then asks a question, like, what was Black’s last move? The answer needs to be inferred from the position, the additional stipulations, if any, and the rules of the game of chess. In other words, every position has to be legal, which means that it was reached by a sequence of legal moves. We have to accept the conditions imposed in the same way that we accept mathematical axioms. Continue reading


A walk on the random side

I find subway maps and their connections fascinating. When I first saw the Tokyo subway map, I played a little mental game. I picked a random station, and followed a series of random routes, and each time I would end up at a different station. So, I thought to myself: if I was a passenger at a random station, and all I had was a list of stations and routes, could I reach any station I wanted without a map? How would I know if I could?

Subway maps and graph connectivity

This problem turns out to be a basic question in graph theory: the problem of graph connectivity. A graph is collections of points called vertices connected by line segments joining them called edges. It asks, given some graph (I shall only look at finite graphs), is there an algorithm to determine whether or not it is connected (in other words, if every pair of vertices is joined by a sequence of edges). If we look back at the subway problem, the parallels between it and the graph connectivity problem are immediately clear. At its essence, a subway map is a graph, made of stations (its vertices) and tracks between stations (its edges).

Continue reading


How Polish cryptographers first broke the unbreakable cipher

In 1926, codebreakers in Room 40 of the Admiralty Old Building, Whitehall—the codebreaking centre of Britain at that time—were monitoring German communications after the first world war, when they started receiving baffling encrypted messages. This was the first time the British encountered messages enciphered by an electromechanical encryption machine first invented in 1918 by the German electrical engineer Arthur Scherbius. He originally marketed them for commercial purposes, such as encrypting communications between banks, but without much success. After some modifications and rebranding, in 1926 (shortly before his death in 1929) he finally found an interested client—the German navy. He called his machine Enigma.

The Enigma cipher would of course become one of the most infamous ciphers in the history of cryptography. We all know stories about Enigma and the people who made a great combined effort to break it during the second world war. But this is an Enigma story you might not have heard. This is the story of how the original Enigma was actually deciphered in Poland in 1932—before the outbreak of the second world war, and long before the celebrated work of Alan Turing, his Bombe, and Bletchley Park.

Continue reading


Celebrating simple mathematical models

In their most uncomplicated form mathematical models are essentially just mathematical descriptions of real-world systems. Stringing together variables and parameters into an equation we can attempt to describe complex behaviours that change with respect to time.

Today mathematical models are used for everything, from predicting exam grades, to the Earth’s climate a hundred years from now. But mathematical models don’t need to be complicated to be useful. Even simple models have the power to reveal insights about a problem, to guide us in the decisions we make and to reveal unexpected consequences of our actions. They even stand up surprisingly well to more sophisticated models, still managing to capture subtle dynamics, with far less computational expense.

I think simple mathematical models are worth celebrating, so here I’m going to discuss three simple but very important models. You might be used to thinking of physics or engineering as the typical subjects in which mathematical models are employed, so let’s turn this on its head and describe some models from sociology and biology.

Survival of the fittest

A key question in sociology and economics is how the size of human populations change over time, and when we talk about something changing with respect to time we use a differential equation to describe it.

The simplest model of population growth comes from 1798: the Malthusian model, \[\frac{\mathrm{d}P}{\mathrm{d}t}=\alpha P,\] where $P(t)$ is the size of the population at time $t$, and $\alpha$ is the constant growth rate of the population. This model argues that the rate at which the population changes over time depends linearly on $P$, the number of people currently able to reproduce or die. If we use the initial condition, that at our commencement point in time the population size is $P(0)=P_0$, then this differential equation can be solved to give \[P(t)=P_0 \mathrm{e}^{\alpha t}.\]

Graph showing global population growing approximately exponentially between 1800 and the presentThis is a very simple conclusion, but is the model any good?

The graph on the right shows how global human population has changed over the last 220 years. The green circles represent the real numbers: data from the United Nations, and the blue line is the mathematical model’s prediction $P(t)$, where $\alpha=0.011$ and $P_0$ is chosen so $P(1850)$ is 1 billion people.

This is a fairly good match; both prediction and data show exponential growth. But what about the future? In Factfulness, the physician and academic Hans Rosling discusses how as countries get richer, citizens tend to have fewer children and growth rate slows, so $\alpha$, and the slope of our graph, should decrease over time.

So is the model perfect? No, but one simple equation, whose solution can be solved used a pen and a single side of A4 paper does a pretty good job of describing how human populations have changed for the last few centuries.

The lynx effect

Now let’s look at a slightly more complicated mathematical model, how the populations of two different animal species interact.

The Canada lynx is a wild cat that lives broadly in Canada and Alaska, with a diet consisting mostly of the snowshoe hare, which is native to the same geographical region. Let’s try to model how the numbers of each animal changes over time.

We use variable $P$ to represent the hare population, and $Q$ for the lynx. The Lotka-Volterra model, from the early 20th century, for how the sizes of these populations change is given below:

Pair of coupled differential equations. The population of hares increases proportional to the current population due to reproduction, and decreases in proportion with the interactions between hares and lynxes. On the other hand the lynx population increases in proportion with their interactions with hares, and decreases in proportion to their population size due to overcrowding.

We can adjust the positive constants, $\alpha$, $\beta$, $\gamma$ and $\delta$, to fit real-life measurements (why shouldn’t we necessarily want $\beta=\delta$?): in modelling we call these parameters.

Hare today… gone tomorrow. Image: Eric Kilby, CC BY-SA 2.0.

Unlike the Malthusian model, solving this system of equations has to be done numerically. The solution is periodic, both the predator and prey populations oscillate, with the predators population trailing slightly behind that of its prey.

Once more, we can compare our model to reality. There is plenty of data recording lynx and hare populations, or to be specific, there is plenty of real historic data about how many lynx and hare pelts were collected by fur traders in the area.

The plot below shows the comparison between the solution of the system and pelt counts from between 1900 to 1920. I’ve found the best-fitting values for the parameters and the initial conditions by using a least squares method (if you want the details, Joseph Mahaffy’s lecture notes from San Diego State University take you through it).

ItGraph showing lynx and hare populations oscillating over a 20 year period. Both populations oscillate with the same frequency, however lynx population maxima follow about two years behind hare population maxima. might not seem intuitive why these populations should oscillate, but let’s have a think. When there is an abundance of tasty hares, there’s more than enough food to go around and the population of the lynxes swells. But a large and greedy population rapidly depletes the hare stash, and food shortages mean a large population can’t be supported for long, and the number of lynxes falls.

Not bad for a couple of differential equations: the model isn’t too hard to understand, it describes this common predator-prey situation well, and it reveals this interesting regular periodicity to how their population sizes change.

Hot topic

Moving on to our final model, we turn our attention to some of the hottest maths in the news. It would be difficult to have missed talk about the ‘R number’ in news reports on the ongoing Covid-19 pandemic. This number, called $R_0$ in the mathematical literature, is the basic reproduction number. Loosely speaking, $R_0$ tells us how many people we expect one person to infect, and it is typically used to refer to the spread of a disease prior to any government interventions to reduce transmission.

$R_0$ comes from the SIR model for the spread of infectious diseases. The key variables in this model come from how we split the initially susceptible population of the country into three groups:

  • $S(t)$, the number of people still susceptible to the disease,
  • $I(t)$, the number of infected people, and
  • $R(t)$, the number of people who have recovered from the disease and developed immunity.

If we make the key assumptions that the total initially susceptible population size does not change over time, $S+I+R=N$, and that the population is completely homogeneous, this then leads directly to a system of nonlinear ordinary differential equations,

Three coupled differential equations. The rate that susceptible people become infected is proportional to the rate of interactions between susceptible and infected individuals. The number of people who recover increases in proportion to the number of infected people.

This is similar to Lotka–Volterra. There, $\delta xy$ represents the growth of the predator population since $xy$ represents interactions. What do you think $\beta SI$ represents here?

We have parameters for the infection rate, $\beta$, and the recovery rate, $\gamma$. The number $R_0$ is the rate at which secondary cases are produced, multiplied by the average infectious period, \[R_0=\beta N \times \frac{1}{\gamma} = \frac{\beta N}{\gamma}.\]

What happens in virus outbreak when no measures are made to contain it? Let’s look at the solution curves for $S$, $I$ and $R$. It’s possible to find these numerically if we choose some parameters. Fitting the model to data from the first wave of Covid-19 in Italy suggests a good match with $\beta N=0.180\,\mathrm{day}^{-1}$, $\gamma=0.037\,\mathrm{day}^{-1}$.

If we start with one infected person, $I=1$, and having everyone initially susceptible to start, $S(0)=N$, applying these parameters to the UK gives us plots for $S$, $I$ and $R$ below on the left.

Plot of numerical solutions to the SIR equations. The number of susceptible individuals decreases quickly in a S-curve, which the number of recovered increases in an S-curve, although at a slower rate. Where these two curves intersect the is a maximum in the number of infected individuals.That’s a peak of over 120,000 infections, 100 days after the first infection. A pretty scary picture, and clear warning that interventions need to be made! If you’re not convinced, at the start of 2020, the UK had just 5,900 critical care beds. Making an assumption that one in ten of those infected in the first wave would need a critical care bed in hospital, then the model predicts that without intervention the NHS would be overwhelmed less than three months after patient zero contracted the disease.

Luckily, on a more positive note, this model also tells us what to do. We have a few different avenues to take action. We could try to change the number of people susceptible $S(t)$, by introducing a vaccine. Or we could try to decrease the average infectious period $1/\gamma$, but this is tricky if we don’t know much about the virus. Possibly the cheapest and easiest solution we could try is to tackle our parameter $\beta$. $\beta$ represents the infection rate; for $R_0$ to go down we need $\beta$ to be smaller. That means for every person who catches the disease we need them to interact with fewer others. The solution? Well perhaps we could consider quarantines, nationwide lockdowns, mask wearing or social distancing… do you see where we’re going here?

For the parameters we’ve chosen $R_0\approx4.9$. For England, with the initial strain in a fully susceptible population, the more sophisticated models from the team at Imperial College London (which were the models used to inform government policy), found $R_0$ at the time to be between 2.5 and 3.3.

But if we have more sophisticated models, why bother with the simple ones at all? Even with modern computing power, simple models are less computationally expensive. This SIR model runs instantaneously on my desktop computer with a few lines of code; the Imperial models use a huge amount of code which needs considerable time to run on a large computer cluster. Even this simple SIR model correctly predicts that $R_0>1$ and hence that the virus spreads, because each person infects more than one other person.

Furthermore, a simple model like this helps the public understand the epidemiological risks of a new virus. The solution curves are intuitive, and the figures arising from the model, like $R_0$, are comprehensible enough to be discussed on primetime news channels. It’s rare for mathematical terminology to seep into public discourse: it takes a very powerful, but simple model to do that.

A happy conclusion

Simple mathematical models are a great gateway into understanding both mathematical concepts and the workings of systems the mathematics seeks to describe. Even very simple models can provide powerful insights; insights which can be gained from more complicated models but only at the expense of elegant equations and quick computations.

Yet there are plenty of real-world problems where a nice simple mathematical model is still lacking. If you’ve played with sand on a beach you’ll know that a collection of grains can flow like a liquid. In this area—known as granular flows—we are still lacking simple models for such a fluid which capture the observed behaviour. Why not consider your favourite physical system and see if you can come up with a simple mathematical model to describe it? Your model might just provide you with a new wealth of insight into what’s actually happening!